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% REFERENCING
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#1\par\bgsection{References}\fi}
% BLACKBOARD BOLD
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\def\ibb #1{\leavevmode\hbox{\kern.3em\vrule
height 1.5ex depth -.1ex width .2pt\kern-.3em\rm#1}}
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% THEOREMS : allow items in proclaim
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\def\QED {\hfill\endgroup\break
\line{\hfill{\vrule height 1.8ex width 1.8ex }\quad}
\vskip 0pt plus100pt}
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\def\nl{\hfill\break}
\def\nll{\hfill\break\vskip 5pt}
% OPERATORS
\def\Aut{{\rm Aut}}
\def\Bar{\overline}
\def\Im{\mathchar"023D\mkern-2mu m} % redefinition!
\def\Nu{{\cal V}}
\def\Order{{\bf O}}
\def\Re{\mathchar"023C\mkern-2mu e} % redefinition!
\def\Set#1#2{#1\lbrace#2#1\rbrace} % \Set\Big#1 to force size of \set
\def\Tr{\tr}
\def\abs #1{{\left\vert#1\right\vert}}
\def\ad#1{{\rm ad}\,#1}
\def\ad{{\rm ad}}
\def\bW{{\widetilde W}}
\def\bom{{\widetilde\om}}
\def\bra #1>{\langle #1\rangle}
\def\bracks #1{\lbrack #1\rbrack}
\def\dim {\mathop{\rm dim}\nolimits}
\def\dom{\mathop{\rm dom}\nolimits}
\def\id{\mathop{\rm id}\nolimits}
\def\ker{{\rm ker}}
\def\ket #1 {\mid#1\rangle}
\def\ketbra #1#2{{\vert#1\rangle\langle#2\vert}}
\def\midbox#1{\qquad\hbox{#1}\qquad} % In displayed formulas
\def\norm #1{\left\Vert #1\right\Vert}
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\def\pr#1{{\rm pr_{#1}}} % projection onto component
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\def\spr{{\rm spr}} % used for spectral radius
\def\stt{\,\vrule\ }
\def\th{\hbox{${}\1{{\rm th}}$}\ } % also in text
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\def\tr{\mathop{\rm tr}\nolimits}
\def\undbar#1{$\underline{\hbox{#1}}$}
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% LETTERS
\def\phi{\varphi} % redefinition!
\def\epsilon{\varepsilon} % redefinition!
\def\om{\omega}
\def\Om{\Omega}
\def\A{{\cal A}} \def\B{{\cal B}} \def\C{{\cal C}} \def\D{{\cal D}}
\def\G{{\cal G}} \def\H{{\cal H}} \def\K{{\cal K}} \def\L{{\cal L}}
\def\M{{\cal M}} \def\N{{\cal N}} \def\R{{\cal R}} \def\T{{\cal T}}
\def\Z{{\cal Z}}
\def\E{{\Ibb E}} \def\Eh{{\hat{\Ibb E}}} \def\F{{\Ibb F}}
\def\DD{{\Ibb D}} \def\P{{\Ibb P}}
\def\Eh{\E_\idty} % = old "E hat"; redefinition for FCD
\def\triple#1{{\def\kind##1{#1} $(\kind\B,\kind\E,\kind\rho)$}}
% use as \triple{#1_1\otimes\tilde#1_2} or \triple{#1}
\def\chain#1{{#1}_\Irs\ }
\def\Aint#1{\A_{\lbrack1,#1\rbrack}}
\def\St{{\cal S}} % state space
\def\Ti{{\cal T}} % translation invariant states
\def\extr{\partial_e} % extreme boundary
% TEXT
\def\cfc{C*-finitely correlated} % "C*" is no longer omitted!!!
\def\cp{completely positive}
\def\pg{purely generated}
\def\dec{decomposition}
\def\vbs{Valence-Bond-Solid}
\def\ti{translation invariant}
\def\ie{i.e.\ } % "." is not end of sentence
\def\eg{e.g.\ }
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% we include a shortened version here.
\let\REF\lstref %%%%%%%%%%% \input fcdr
\REF AF \AFri \par
\REF AKLT \AKLT \par
\REF BR \BraRo \par
\REF FNWa \FCS \par
\REF FNWb \FCP \par
\REF LOS \LIN \par
\REF Pou \POU \par
\REF We1 \STEXa \par
\REF We2 \STEXb \par
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% Here begins the paper proper
% File: FCD.tex
% \let\draft1 % suppress \remark s
\font\BF=cmbx10 scaled \magstep 3
\line{\hfill \tt Preprint KUL-TF-92/23}
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\centerline{\BF Abundance of Translation Invariant Pure States}
\vskip 10pt
\centerline{\BF on Quantum Spin Chains}
\vskip 40pt plus40pt
\centerline{
M. Fannes$\1{1,2}$, B. Nachtergaele$\1{3}$, and R.F. Werner$\14$}
\vskip 12pt
\centerline{\tt fgbda20@blekul11.bitnet\quad bxn@math.princeton.edu
\quad reinwer@dosuni1.bitnet}
\vskip 80pt plus80pt
\noindent {\bf Abstract}\hfill\break
We construct a set of \ti\ pure states of a quantum spin chain,
which is w*-dense in the set of all \ti\ states of the chain.
Each of the approximating states has exponential decay of
correlations, and is the unique ground state of a finite range
Hamiltonian with a spectral gap above the ground state energy.
\vfootnote1
{Inst. Theor. Fysica, Universiteit Leuven, B-3001 Leuven, Belgium}
\vfootnote2
{Bevoegdverklaard Navorser, N.F.W.O. Belgium}
\vfootnote3
{Dept. of Physics, Princeton University, NJ-08544-708, USA;\nl
\vrule width 0pt \qquad on leave from Universiteit Leuven, B-3001
Leuven, Belgium}
\vfootnote4
{Fachbereich Physik, Universit\"at Osnabr\"uck, Pf. 4469,
Osnabr\"uck, Germany}
\vfill\eject
%\beginsection Introduction
It has long been noted that quantum statistical mechanics is in many
ways more difficult than statistical mechanics based on a classical
microscopic theory. For example, the thermodynamics of one-dimensional
classical spin systems can be solved completely using the transfer
matrix technique, whereas in the quantum case even very basic
questions about the ground states of systems with nearest neighbour
interactions remain unanswered. The main difficulty in the study of
quantum spin chains is perhaps constructing \ti\ states with given
local expectations. The fundamental problem with such constructions is
the fact that families of quantum states defined on overlapping
subintervals of the chain fail to have common extensions to the whole
chain algebra \tref\STEXb, whereas for classical chains common
extensions always exist.
Consequently, a global state of minimal energy density
cannot be found by varying the state on local subalgebras. Similar
problems arise in the classical case only for dimensions two and
higher, where Bell-type inequalities associated to cycles in the
lattice \tref\STEXa\ constitute obstructions to such state
extensions, also known as ``frustration''.
In this note we demonstrate a new property of states on
one-dimensional quantum systems, which highlights another aspect of
the higher complexity of these systems in comparison with classical
ones.
To be specific we shall consider a system of ``spins'' localized
on the sites of the one-dimensional lattice $\Ir$, each of which is
described by the same observable algebra $\A_i\equiv\A$, which we
take as the algebra $\M_d$ of $d\times d$-matrices. For a finite
subset $\Lambda\subset\Ir$ the observable algebra is
$\bigotimes_{i\in\Lambda}\A_i$, and for the entire chain the
observable algebra is the inductive limit of these algebras, i.e.\
the norm closure $\chain\A$ of $\bigcup_N\A_{\bracks{-N,N}}$.
Clearly, there is an action $(\tau_x)_{x\in\Ir}$ of the
lattice translations by automorphisms of $\chain\A$. For convenience
we will often write $\tau$ instead of $\tau_1$.
By $\St$, or $\St(\chain\A)$ for emphasis, we will denote the set of
states on $\chain\A$. This is a compact convex set with the
w*-topology. By definition, a net of states $\omega_\alpha\in\St$
converges in this topology to $\omega\in\St$ if
$\omega_\alpha(A)\to\omega(A)$ for all $A\in\chain\A$. The \ti\
states will be denoted by
$\Ti=\set{\omega\in\St\stt \omega\circ\tau=\omega}$. This is a
w*-closed subset of $\St$. For any convex set we will denote by
$\extr\K$ its extreme boundary, i.e.\ the points $\omega\in\K$ which
cannot be represented as a convex combination
$\omega=\lambda\omega_1+(1-\lambda)\omega_2$ with $0
=\bra\chi_m,\chi_n>\, \bra\phi_{m+1},X\phi_{n+1}>
\quad.$$
Since $\abs{\bra\chi_m,\chi_n>}<1$ for $n\neq m$, multiples of
$\idty$ are the only fixed vectors of $\EhV$, and one easily
checks that its unique invariant state is the normalized trace.
Its peripheral eigenvalues are exactly $\exp(2\pi il/k)$,
$l=0,\ldots,k-1$.
We show the density of $\Nu_a$ by an analyticity argument.
Let $V_1\in\Nu_a$, and $V\in\Nu\setminus\Nu_a$.
Then we may find a hermitian $\Theta\in\B(\H\otimes\K)$ such
that $V_1=\exp(i\Theta)V$, and set
$V_\lambda=\exp(i\lambda\Theta)V$. For real $\lambda$ we
therefore have $V_\lambda\in\Nu$, and
$\lambda\mapsto\Eh\1{V_\lambda}(X)
=V\1*\exp(-i\lambda\Theta)X\exp(i\lambda\Theta)V$
extends to an entire analytic function for all $X$. Let
$$ f(\lambda,z)
=(z-1)\1{-1}\det\bigl(\Eh\1{V_\lambda}-z\id\bigr)
\quad.$$
Then $f$ is a polynomial in $z$, because all $\Eh\1V$ share the
fixed point $\idty$. Moreover, $V_\lambda\in\Nu_a$ if and only if
$f(\lambda,1)\neq0$. Clearly, $\lambda\mapsto f(\lambda,1)$ is
entire and not constant, because $f(1,1)\neq0=f(0,1)$. Hence
$\lambda=0$ is an isolated zero of $f(\cdot,1)$, and
$V_\lambda\in\Nu_a$ for small $\lambda$.
Consequently, $\Nu_a\subset\Nu$ is dense.
To show the density of $\Nu_b$ we show that the complement of this set
is contained in a finite union of submanifolds $\Nu_s\subset\Nu$ of
strictly smaller dimension.
Here we consider $\Nu$ as a homogeneous space
of the unitary group $\U(kd)$ of $\B(\H\otimes\K)$: if
$V_1:\K\to\H\otimes\K$ is any isometry, we can obtain all
others by multiplying $V_1$ from the left by a unitary in $\U(kd)$.
The fixed subgroup of $V_1$ consists of those unitaries, which are
the identity on the range of $V_1$ and arbitrary on its orthogonal
complement. Hence $\Nu\cong\U(kd)/\U(kd-k)$ and the real manifold
dimension of $\Nu$ is $\dim\Nu=(kd)\12-(k(d-1))\12=k\12(2d-1)$.
Suppose now that $\Eh\1V$ has an invariant state $\rho$ with support
projection $S1$, and let $V_1$ be
any isometry such that $\zeta$ is not an eigenvalue of $\Eh\1{V_1}$,
and consider the function $f$ defined above. Then $f(0,\zeta)=0$, and
$f(\zeta,1)\neq0$, so by analyticity we conclude that
$f(\lambda,\zeta)\neq0$ for small $\lambda$. Hence
$\Nu_c\cap\Nu_b\cap\Nu_a$ is dense in $\Nu$.
\QED
\proof{ of the Theorem:}
\def\step#1{\par\noindent{\bf Step #1:}\hfill\break}%
\def\Cd#1{\ifx#11{\Cx\1d}\else(\Cx\1d)\1{\otimes\,#1}\fi}%
\def\omps{\omega_{\ell,\psi}}%
\def\lm{\ell-1}%
Our strategy of proof is the following: we argue in step 1 that
any \ti\ $\omega$ can be approximated by pure product states with
respect to a sufficiently coarse partitioning of $\Ir$ into equal
intervals. Since we want to approximate by {\it \ti} states, we take
the averages over such states with respect to translations, and show
that such averages still approximate $\omega$.
As the degree of approximation is improved the averaging makes the
approximating states convex combinations of more and more states,
hence less and less pure. The main problem is thus to replace these
impure states by w*-close pure ones.
In step 3 we show that the averaged states are of the form
$\omega\1V$. In step 5 we then use the Lemma to deform the $V$
obtained in step 3 into $V'$ such that $\omega\1{V'}$ becomes pure by
virtue of the Proposition. This concludes the proof, but in order to
conclude in step 5 that $\omega\1{V'}$ is close to $\omega\1V$ we need
the continuity of $\omega\1V$ with respect to $V$, which holds only
for $V\in\Nu_a$. We show in step 4 that the $V$ from step 3 is in
$\Nu_a$, provided that a non-degeneracy property of the averaged
states holds, which we establish without loss of generality in step
2.
\step1
For $\ell\in\Nl$ and $\psi\in\Cd\ell$ a unit vector, let $\omega_\psi$
denote the product state on the regrouped chain
$\chain{(\A\1{\otimes\ell})}$ formed from the pure state on
$\A\1{\otimes\ell}$ given by $\psi$. Then we consider the states
$$ \omps
={1\over\ell}\sum_{p=0}\1{\ell-1}\ \omega_\psi\circ\tau_p
\quad.\eqno(3)$$
We claim that states of this special form are w*-dense in $\Ti$. To
see this, let $\omega$ be \ti. For any even $\ell\equiv2n$, we can
find a unit vector $\psi\in\Cd{\ell}$ with the property that
$\om(A\1{(n)})=\bra\psi\mid\idty\1{\otimes n}\otimes A\1{(n)}\psi>
=\bra\psi\mid A\1{(n)}\otimes\idty\1{\otimes n}\psi>$
for all $A\1{(n)}\in\Aint n\cong\M_d\1{\otimes n}$.
This follows from the GNS-construction for the state
$\om\rstr\Aint n$.
For any $A\in \Aint m$ one immediately gets from the definition of
$\omps$ that
$$\eqalign{\omps(A)
&={1\over \ell}\sum_{p=0}\1{\ell-1}\om_\psi(\tau_p(A))\cr
&={1\over 2n }\left\{2(n-m+1)\om(A)
+\sum_{p=n-m+1}\1{n-1} \om_\psi\circ\tau_p(A)
+\sum_{p=\ell-m+1}\1{\ell-1}\om_\psi\circ\tau_p(A)\right\}\cr
}$$
and hence
$$ \abs{\om(A)-\omps(A)}
\leq {2(m-1)\over n}\norm{A}
\quad.$$
Hence for each fixed $A$ we have
$\lim_{n\to\infty}\om_{2n,\psi}(A)=\omega(A)$, and the states given
by equation (3) are w*-dense in $\Ti$.
\step2
We claim that the subset of states $\omps$ such that $\psi$ is not of
the form $\psi_p\otimes\psi_{\ell-p}$ for any $p$ with $0

$.
\item{}}
Because $\norm\psi=1$, $V$ is an isometry.
We omit the straightforward verification that $\rho$ is invariant
under $\Eh\1V$.
%
% It is easy to check that $\rho$ is indeed an invariant state of
% $\Eh\1V$:
% $$\eqalign{
% \rho(\Eh\1V(\bigoplus_{p=0}\1{\ell-1}B_p))
% &=\rho(\bra\psi\mid\idty\otimes B_{\ell-1}\psi>
% \oplus \bigoplus_{p=0}\1{\ell-2}\idty\otimes B_p)\cr
% &={1\over \ell}\set{\bra\psi\mid\idty\otimes B_{\ell-1}\psi>
% \bra\psi\mid\psi>
% +\sum_{p=0}\1{\ell-2}\bra\psi\mid\idty\otimes B_p\psi>}\cr
% &=\rho(\bigoplus_{p=0}\1{\ell-1}B_p)\cr
% }$$
%
For any $A\in\M_d$, the operator $\E\1V_A$ then leaves the
block-diagonal algebra $\bigoplus_{p=0}\1{\ell-1}(\M_d)\1{\otimes p}$
invariant and it is given by:
$$\eqalignno{
\E\1V_A(\bigoplus_{p=0}\1{\ell-1}B_p)
&=V\1*A\otimes\bigoplus_{p=0}\1{\ell-1}B_p V\cr
&=\bra\psi\mid A\otimes B_{\ell-1}\psi>
\oplus \bigoplus_{p=0}\1{\ell-2}A\otimes B_p
\quad.\cr}$$
Iterating this equation $\ell$ times we find for
$A_1,\ldots A_\ell\in\M_d$:
$$ (\E\1V_{A_1}\cdots\E\1V_{A_\ell})
\bigl(\bigoplus_{p=0}\1{\ell-1}B_p\bigr)
=\bigoplus_{p=0}\1{\ell-1} A_1\otimes\cdots A_p
\bra\psi, A_{p+1}\otimes\cdots A_\ell\otimes B_p \psi>
\quad.$$
>From this equation it can be checked easily that $\om\1V=\omps$.
\step4
We now show that, provided $\psi$
is not factorizable as $\psi_p\otimes\psi_{\ell-p}$, the $V$
constructed in the previous step satisfies condition (a) of the
Lemma.
Any operator $B\in\M_k$ can be written in block matrix form with
respect to the decomposition
$\Cx\1k =\bigoplus_{p=0}\1{\ell-1}\Cd p$.
Thus the entry $B_{ij}$ of $B$ is an operator from
$\Cd i$ to $\Cd j$, or a $d\1i\times d\1j$-matrix.
$\Eh\1V$ then acts like
$$\eqalign{
\Eh\1V(B)
&=\Eh\1V\pmatrix{B_{00}&B_{01}&\cdots&B_{0,\lm}\cr
\vdots& & & \vdots\cr
B_{\lm,0}&B_{\lm,1}&\cdots&B_{\lm,\lm}\cr} \cr
\hbox{\vbox to 5pt{}}\cr
&=\pmatrix{
\bra\psi\mid\idty\otimes B_{\lm,\lm}\psi>
&\langle\psi\mid\idty\otimes B_{\lm,0}&\cdots
&\langle\psi\mid\idty\otimes B_{\lm,\ell-2}\cr
\idty\otimes B_{0,\lm}\mid\psi\rangle
&\idty\otimes B_{00}&\idty\otimes B_{01}&\cdots\cr
\vdots& & & \vdots\cr
\idty\otimes B_{\ell-2,\lm}\mid\psi\rangle
&\idty\otimes B_{\ell-2,0}&\cdots&
\idty\otimes B_{\ell-2,\ell-2}\cr}
\quad.}$$
Thus the $(i,j)$-entry of $(\Eh\1V)\1n(B)$ depends only on $B_{i'j'}$
with $i-i'\equiv j-j'\equiv n\ {\rm mod} \ell$. A fixed point $B$ is
uniquely determined by the entries $B_{m0}, m=0,\ldots\ell-1$, and the
fixed point condition can be evaluated separately for each $m$. For
$m=0$, $B_{00}$ is just a scalar. Thus the fixed point $B=\idty$
is the unique one for $m=0$, up to a factor.
Now suppose $B_{m0}\neq0$ for some fixed point $B$. Since
$B_{m0}:\Cx\to\Cd m$, this entry is given uniquely in terms of the
vector $\phi_m=B_{m0}1$. Iterating the condition $\Eh\1V(B)=B$ we
obtain
$$\eqalign{
B_{\lm,\lm-m}\xi_{\lm-m} &=\xi_{\lm-m}\otimes\phi_m\cr
B_{0,\ell-m} \xi_{\ell-m}
&=\bra\psi,\xi_{\ell-m}\otimes\phi_m>\cr
\bra\xi_{m-1},B_{m-1,\lm}\xi_{\lm}>
&=\bra\xi_{m-1}\otimes\psi,\xi_{\lm} \otimes\phi_m>\cr
\bra\xi_{m},B_{m0}\xi_0>
&=\bra\xi_m\otimes\psi,\psi\otimes\phi_m>\xi_0 \cr
&=\bra\xi_m,\phi_m> \xi_0
\quad, \cr}$$
where in every equation $\xi_j$ denotes an arbitrary vector in
$\Cd j$.
Hence with $\norm{\psi}=1$ we get the condition
$\bra\phi_m\otimes\psi,\psi\otimes\phi_m>
=\norm{\phi_m}\12\norm{\psi}\12$,
which by the Cauchy-Schwartz inequality implies
$$ \phi_m\otimes\psi=\psi\otimes\phi_m
\quad.$$
Unless $\phi_m=B_{m0}1=0$, this implies a factorization of $\psi$
\tref\STEXb\ of the type we have excluded by assumption. Hence the
fixed point $\idty$ is unique up to a factor.
\step5
Combining the steps so far, we have shown that the set of states
$\om\1V$ with $V$ satisfying the hypothesis (a) of the Lemma (\ie
$V\in\Nu_a$) is w*-dense in $\Ti$. It is evident that $\Eh\1V$ depends
continuously on $V$. By analytic functional calculus the fixed point
$\rho$ depends continuously on $\Eh\1V$, where it is unique. Hence
$\rho=\rho\1V$ is a continuous function of $V$ for $V\in\Nu_a$. For any
finite $n$, and any local observable $A=A_1\otimes\cdots A_m\in\Aint
m$ the function $V\mapsto
\om\1V(A)=\rho\1V\circ\E\1V_{A_1}\cdots\E\1V_{A_m}(\idty)$ is
continuous. By taking linear combinations and norm limits, this
result carries over to arbitrary $A\in\chain\A$. Thus
$V\mapsto\om\1V$ is w*-continuous on $\Nu_a$. By the Lemma we may
approximate any $V\in\Nu_a$ by elements $V'\in\Nu_{abc}$, and by the
Proposition the states $\om\1{V'}$ are pure. Hence the pure states
of this form are w*-dense in $\Ti$.
\QED
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% second run of FCDR.tex :
\let\REF\doref %%%%%%%%%%% \input fcdr
\ACKNOW
B.N. is partially supported by NSF Grant \# PHY-8912069.
R.F.W is supported by a fellowship from the DFG in Bonn, and also
acknowledges a travel grant to visit Princeton, where a part of this
work was carried out.
\REF AF \AFri \Jref
L. Accardi, and A. Frigerio "Markovian Cocycles"
Proc.R.Ir.Acad. @83A{(2)}(1983) 251--263
\REF AKLT \AKLT \Jref
I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki
"Valence bond ground states in isotropic quantum
antiferromagnets"
Commun.Math.Phys. @115(1988) 477--528
\REF BR \BraRo \Bref
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\bye